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  1. 2 Open Mathematics

Autorzy help

  1. 1 Athorne C.

  2. 1 Chen T.

  3. 1 England M.

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  1. 2 2012

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The associated map of the nonabelian Gauss-Manin connection

100%
Chen T.
Open Mathematics
|
2012
|
tom 10
|
nr 4
1407-1421
EN
The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow. We write down explicitly the vector fields of the isomonodromy flow and calculate its induced vector fields on the associated graded space of the nonabelian Hogde filtration. The result turns out to be intimately related to the quadratic part of the Hitchin map.
2
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Generalised elliptic functions

64%
England M., Athorne C.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1655-1672
EN
We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ℘-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study. The first approach discussed sees the functions defined as logarithmic derivatives of the σ-function, a modified Riemann θ-function. We can make use of known properties of the σ-function to derive power series expansions and in turn the properties mentioned above. This approach has been extended to a wide range of non hyperelliptic and higher genus curves and an overview of recent results is given. The second approach defines the functions algebraically, after first modifying the curve into its equivariant form. This approach allows the use of representation theory to derive a range of results at lower computational cost. We discuss the development of this theory for hyperelliptic curves and how it may be extended in the future. We consider how the two approaches may be combined, giving the explicit mappings for the genus 3 hyperelliptic theory. We consider the problem of generating bases of the functions and how these decompose when viewed in the equivariant form.
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